No other answer seems to mention this, so I'll add it.
Throughout most of the run, the Doctor is treated (as much as possible) the same as any other member of the crew. As critical as he is, he's allowed to continue to expand his program, even though it's been shown to be dangerous (e.g., S03E18, Darkling, where he develops a split personality).
In addition to singing and holophotography, he's allowed to:
- Create a holographic family (S03E22, Real Life)
- Daydream (S06E04, Tinker Tenor Doctor Spy)
- Participate in away missions (note that his holo-emitter is 29th century technology, and as far as we know cannot be replicated, so he risks both that technology and his program in these cases).
He's even granted permission to remain behind on a planet, leaving Voyager without his services (S06E13, Virtuoso), and to travel to the Alpha Quadrant, using a communication link that may not be entirely reliable, risking the permanent loss of his services, to treat his creator (S06E24, Life Line)
We've been reminded, on occasion, that he is a program, such as when the crew of the Equinox removed his ethical subroutines, and he was perfectly happy to experiment on Seven of Nine, even if it would permanently injure her (S05E26 - S06E01, Equinox).
We've also seen that monkeying around with his memories can cause problems (S05E11, Latent Image, where a patient died because of a triaging choice, but he blamed himself and became unable to function). And, the ultimate solution, in that story? Treat him like a biological crewmember, not a program; help him work through the issue until he can get back to work, where he'll probably be an even better doctor for the experience.
So, when he's in the Alpha Quadrant for a month, he's not replaced by a backup. As others have noted, that would leave them in the position of having two programs with unique experiences, and with potential difficulty in merging the programs together again. And, since he has been accepted as a member of the crew, it would be unthinkable to simply delete one copy.
Would a backup copy (assuming they exist, as indicated by "Living Witness") of the Doctor have been used if he had actually left the ship in "Virtuoso"? We simply don't know. He decided not to leave before such an eventuality would have been necessary. I will note that, apparently, he would have taken the one-of-a-kind mobile emitter with him if he left.
TL;DR - The fact that the Doctor is accepted and treated as a member of the crew, not merely a holoprogram, may ethically limit the uses for his program as much as it's limited by the technology on board.